64 Digital Control Systems

Figure 2.36. Nyquist plot for the nominal open loop transfer function and the real open loop transfer function in presence of uncertainties and parameters variations (H_OL and H’_OL are stable)

76 Digital Control Systems

where S(z^-1) and R(z^-1) are computed on the basis of Equation 2.6.3 for the nominal values of A(z^-1) and B(z^-1).

In other words, the magnitude of S^-1_yp (e^-jω ) function (evaluated in dB units), obtained by symmetry from S_yp (e^-jω ) (see Figure 2.32), gives, at each frequency, a sufficient condition for the accepted difference (computed as the Euclidian distance) between the real open loop transfer function and the nominal open loop transfer function, in order to guarantee the stability of the closed loop.
This tolerance is higher at low frequencies (see Figure 2.32) where the gain of the open loop system is high (especially when an integrator is included), and it has a minimum value at the frequency (or frequencies) where S^-1_yp (e^-jω ) reaches its minimum (= ∆M), that is the frequency where S_yp (e^-jω ) has the maximum value.
It is necessary to ensure that at these frequencies the plant model variations are compatible with the obtained modulus margin. If this is not the case, the solution is to provide a more accurate model, or to modify the specifications in order to maintain the closed loop stability.
Equation 2.6.6 expresses a robustness condition in terms of open loop transfer function variations (controller + plant). It is interesting to express this robustness condition in terms of the plant model variations only. Note that Equation 2.6.6 can be further expressed as

where B(z^-1)/A(z^-1) corresponds to the nominal plant transfer function.
Multiplying by |S(z^-1)/R(z^-1)| both sides of Equation 2.6.7 one gets the condition

By plotting the inverse of the input sensitivity function magnitude, sufficient conditions for tolerated (additive) variations (or uncertainties) of the plant transfer function are obtained. The inverse of the magnitude of the input sensitivity function is symmetric to the input sensitivity function magnitude in dB units with respect to the axis at 0 dB (see Figure 2.37).
As plant model uncertainties at high frequencies are often present, one must verify that the maximum of |S_up (e^-jω )| at high frequencies is small. On the other
Computer Control Systems 77
hand, the input sensitivity function Sup is an effective image of the actuator stress in the frequency domain when disturbances act on the system. The physical characteristics of the actuator often impose a bound on actuator stress at high frequencies, and an upper bound of the maximum of |Sup (e-j ω )| at these frequencies should be imposed.
Notice that (from Equation 2.6.8) the admitted tolerances (neglecting the term 1/|R(z^-1)|) depend to a large extent upon the relation between the open loop system poles (defined by A(z^-1)) and the desired closed loop poles (defined by P(z^-1)).
In order to understand this phenomenon in greater detail, Figure 2.38 shows the S_up(z^-1) magnitude functions for a plant model characterized by A(z^-1)= 1 – 0.8 z^-1; B(z^-1)= z^-1 and for two different desired closed loop system characteristic polynomials: P₁(z^-1)=1-0.6 z^-1 and P₂(z^-1)=1-0.3 z^-1 (the controller includes an integrator). Note that P₂(z^-1) corresponds to a closed loop system faster than the one specified by P₁(z^-1), and both closed loop systems are faster than the plant (open loop system).
The |S_up(z^-1)| maximum for P₂(z^-1) is greater than for P₁(z-1), and then the inverse of |S_up(z^-1)| will be smaller. As a consequence, the accepted tolerances for the frequency response variations (especially at high frequencies) are smaller in the case of P₂(z^-1) with respect to the case of desired closed loop performances imposed by P₁(z^-1).

(Tolerance to additive uncertainties)